Foundations of Data Science

6
6
1
8
1 5
8
4
5
5
Graph with boundary vertices
dark and boundary conditions
specified.
3
5
Values of harmonic function
satisfying boundary conditions
where the edge weights at
each vertex are equal
3
Figure 5.6: Graph illustrating an harmonic function.
Example: Convert an electrical network with conductances c xy to a weighted, undirected
graph with probabilities p xy . Let f be a function satisfying fP = f where P is the matrix
of probabilities. It follows that the function g x = fx
c x
is harmonic.
g x = fx
c x
= 1
∑
c x
y
∑
= 1 c
c x
f
xy
y c y
y
f y p yx = 1
= ∑ y
f y
c y
c xy
c x
c x
∑
y
= ∑ y
f y
c yx
c y
g y p xy
A harmonic function on a connected graph takes on its maximum and minimum on
the boundary. Suppose the maximum does not occur on the boundary. Let S be the
set of interior vertices at which the maximum value is attained. Since S contains no
boundary vertices, ¯S is nonempty. Connectedness implies that there is at least one edge
(x, y) with x ∈ S and y ∈ ¯S. The value of the function at x is the weighted average
of the value at its neighbors, all of which are less than or equal to the value at x and
the value at y is strictly less, a contradiction. The proof for the minimum value is identical.
There is at most one harmonic function satisfying a given set of equations and boundary
conditions. For suppose there were two solutions, f(x) and g(x). The difference of two
solutions is itself harmonic. Since h(x) = f(x)−g(x) is harmonic and has value zero on the
boundary, by the min and max principles it has value zero everywhere. Thus f(x) = g(x).
The analogy between electrical networks and random walks
161